Virtual photo-mask critical-dimension measurement

ABSTRACT

A technique for reconstructing a mask pattern corresponding to a photo-mask using a target mask pattern (which excludes defects) and an image of at least a portion of the photo-mask is described. This image may be an optical inspection image of the photo-mask that is determined using inspection optics which includes an optical path, and the reconstructed mask pattern may include additional spatial frequencies than the image. Furthermore, the reconstructed mask pattern may be reconstructed based on a characteristic of the optical path (such as an optical bandwidth of the optical path) using a constrained inverse optical calculation in which there are a finite number of discrete feature widths allowed in the reconstructed mask pattern, and where a given feature has a constant feature width. Consequently, the features in the reconstructed mask pattern may each have the constant feature width, such as an average critical dimension of the reconstructed mask pattern.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to techniques for inspecting andqualifying a photo-mask. More specifically, the invention relates to atechnique for determining whether or not a photo-mask is acceptable byreconstructing a mask pattern of the photo-mask from a lower resolutionimage.

2. Related Art

Photolithography is a widely used technology for producing integratedcircuits. In this technique, a light source illuminates a photo-mask.The resulting spatially varying light pattern is projected on to aphotoresist layer on a semiconductor wafer by an optical system(referred to as an ‘exposure tool’). By developing the 3-dimensionalpattern produced in this photoresist layer, a layer in the integratedcircuit is created. Furthermore, because there are often multiple layersin a typical integrated circuit, these operations may be repeated usingseveral photo-masks to produce a product wafer.

Unfortunately, as dimensions in integrated circuits steadily become asmaller fraction of the wavelength of the light used to expose images ofthe photo-mask onto the wafer, the structures in or on the idealphoto-mask (which corresponds to a ‘target mask pattern’) and/or thephysical structures in or on the actual photo-mask (which corresponds toa fabricated ‘mask pattern’) bear less and less resemblance to thedesired or target pattern at the wafer. These differences between thetarget mask pattern and the target pattern are used to compensate forthe diffraction and proximity effects that occur when light istransmitted through the optics of the exposure tool and is convertedinto the 3-dimensional pattern in the photoresist.

From a photo-mask or reticle manufacturing standpoint, the increasingdissimilarity between the photo-mask and the corresponding waferpatterns creates a broad new class of problems in photo-mask inspectionand qualification. For example, if a defect in a photo-mask is detected,it is often unclear what impact this defect will have on the finalpattern in the photoresist. In addition, photo-mask inspection devicesoften have a different numerical aperture, a different illuminationconfiguration (or ‘source aperture,’ which is also referred to as a‘source pattern’), and even different light wavelength(s) than thoseused in the exposure tool. As a consequence, the image measured by anoptical photo-mask inspection tool is often neither a perfect replica ofthe physical photo-mask nor the pattern (i.e., the aerial image) thatwill be exposed onto the wafer.

One existing approach to these challenges is to use a computer tosimulate the resulting wafer pattern based on the optical inspectionimages of the photo-mask. By comparing simulations of wafer patternscorresponding to the ideal photo-mask (i.e., the target mask pattern)and those associated with an estimate of the actual photo-maskcorresponding to the optical inspection images of the photo-mask, thesignificance of the defect may be determined. However, since the opticalinspection images of the photo-mask may not be an accuraterepresentation of the actual photo-mask, errors may be introduced whensimulating wafer patterns, and thus, when trying to identify or classifydefects. This may further complicate photo-mask inspection andqualification.

Alternatively, a higher-resolution image of the photo-mask than what cantypically be obtained using the optical photo-mask inspection tool maybe used in the simulations. For example, a spatial variation of amagnitude of the transmittance of the photo-mask may be determined usinga scanning electron microscope (SEM). The resulting high-resolutionimage of the photo-mask (which is sometimes referred to as a‘critical-dimension scanning-electron-microscope’ or ‘CD-SEM’ image) canprovide a more accurate representation of the physical photo-mask thanan optical inspection image.

However, measuring a CD-SEM image can be time consuming and complicated.In particular, the electron beam in an SEM often leads to charging ofthe photo-mask, which can attract dirt or contamination. Furthermore,subsequent cleaning of the photo-mask may produce additional defects inthe photo-mask that are not included in the CD-SEM image and, thus, willnot be included in the simulations. Therefore, even though a CD-SEMimage can provide a more accurate representation of the originalphysical photo-mask, errors may still be introduced when simulatingwafer patterns, and thus, when trying to identify or classify defects.

Hence, what is needed is a photo-mask inspection and qualificationtechnique that overcomes the problems listed above.

SUMMARY OF THE INVENTION

The present disclosure relates to a computer system for determining areconstructed mask pattern. During operation, the computer systemreceives a target mask pattern and an image of at least a portion of aphoto-mask corresponding to a mask pattern, where the image isdetermined using inspection optics that includes an optical path. Then,the computer system reconstructs the mask pattern from the image basedon a characteristic of the optical path and the target mask pattern.Furthermore, the mask pattern is reconstructed using a constrainedinverse optical calculation in which there are a finite number ofdiscrete feature widths allowed in the reconstructed mask pattern, andwhere a given feature has a constant feature width.

Note that the reconstructed mask pattern may be characterized byadditional spatial frequencies than the image. Moreover, note that thespatial frequencies in the image may be approximately greater than orequal to an average critical dimension of the target mask pattern.

In some embodiments, the characteristic includes a range of wavelengthstransmitted via the optical path.

Moreover, the features in the reconstructed mask pattern may each havethe constant feature width. For example, the constant feature width maybe an average critical dimension of the reconstructed mask pattern.Alternatively, primary features in the reconstructed mask pattern mayeach have the constant feature width, and sub-resolution assist featuresin the reconstructed mask pattern may each have another constant featurewidth, which is different than the constant feature width.

In some embodiments, the computer system compares the average criticaldimension of the reconstructed mask pattern to the average criticaldimension of the target mask pattern to determine a critical-dimensionbias of the photo-mask. Furthermore, the computer system may determinean acceptance condition of the photo-mask based on thecritical-dimension bias.

Note that the average critical dimension may be associated with aminimum of a cost function in the inverse optical calculation, where thecost function corresponds to a difference between the reconstructed maskpattern and the target mask pattern.

In the constrained inverse optical calculation, the image may be at animage plane of a model of the optical path and the mask pattern may beat an object plane of the model of the optical path. Furthermore, byconstraining the inverse optical calculation, a convergence time of theinverse optical calculation may be faster than that of an unconstrainedinverse optical calculation.

Another embodiment provides a method including at least some of theabove-described operations.

Another embodiment provides a computer-program product for use inconjunction with the computer system.

INCORPORATION BY REFERENCE

All publications, patents, and patent applications mentioned in thisspecification are herein incorporated by reference to the same extent asif each individual publication, patent, or patent application wasspecifically and individually indicated to be incorporated by reference.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth with particularity inthe appended claims. A better understanding of the features andadvantages of the present invention will be obtained by reference to thefollowing detailed description that sets forth illustrative embodiments,in which the principles of the invention are utilized, and theaccompanying drawings of which:

FIG. 1A is a flow chart illustrating a process for determining areconstructed mask pattern in accordance with an embodiment of thepresent invention.

FIG. 1B is a flow chart illustrating a process for determining areconstructed mask pattern in accordance with an embodiment of thepresent invention.

FIG. 2A illustrates a reconstructed mask pattern in accordance with anembodiment of the present invention.

FIG. 2B illustrates a reconstructed mask pattern in accordance with anembodiment of the present invention.

FIG. 3 is a block diagram illustrating a photo-mask inspection andqualification system in accordance with an embodiment of the presentinvention.

FIG. 4 is a block diagram illustrating an inverse optical calculation inaccordance with an embodiment of the present invention.

FIG. 5 is a block diagram illustrating a forward optical calculation inaccordance with an embodiment of the present invention.

FIG. 6 illustrates a mask pattern and corresponding level-set functionsin accordance with an embodiment of the present invention.

FIG. 7 is a block diagram illustrating a computer system in accordancewith an embodiment of the present invention.

FIG. 8 is a block diagram illustrating an image data structure inaccordance with an embodiment of the present invention.

Note that like reference numerals refer to corresponding partsthroughout the drawings. Moreover, multiple instances of the same partare designated by a common prefix separated from an instance number by adash.

DETAILED DESCRIPTION OF THE INVENTION

The following description is presented to enable any person skilled inthe art to make and use the invention, and is provided in the context ofa particular application and its requirements. Various modifications tothe disclosed embodiments will be readily apparent to those skilled inthe art, and the general principles defined herein may be applied toother embodiments and applications without departing from the spirit andscope of the present invention. Thus, the present invention is notintended to be limited to the embodiments shown, but is to be accordedthe widest scope consistent with the principles and features disclosedherein.

Embodiments of a computer system, a method, and a computer programproduct (i.e., software) for use with the computer system are described.These devices and processes may be used to reconstruct a mask patterncorresponding to a photo-mask using a target mask pattern (whichexcludes defects) and an image of at least a portion of the photo-mask.This image may be an optical inspection image of the photo-mask that isdetermined using inspection optics which includes an optical path, andthe reconstructed mask pattern may include additional spatialfrequencies than the image. Furthermore, the reconstructed mask patternmay be reconstructed based on a characteristic of the optical path (suchas an optical bandwidth of the optical path) using a constrained inverseoptical calculation in which there are a finite number of discretefeature widths allowed in the reconstructed mask pattern, and where agiven feature has a constant feature width. Consequently, the featuresin the reconstructed mask pattern may each have the constant featurewidth, such as an average critical dimension of the reconstructed maskpattern.

By reconstructing the mask pattern, this calculation technique mayfacilitate improved photo-mask inspection and qualification. Forexample, the average critical dimension can be used to determine acritical-dimension bias of the photo-mask (relative to an averagecritical dimension of the target mask pattern) and, thus, to determinean acceptance condition of the photo-mask. Furthermore, thereconstructed mask pattern can be determined without measuring acritical-dimension scanning-electron-microscope (CD-SEM) image, andtherefore may be less time consuming and less complicated than otherphoto-mask inspection techniques. Additionally, the calculationtechnique may not introduce additional defects in the photo-mask, whichmay improve photo-mask and wafer yield, thereby reducing overallphotolithography cost and the time to market of integrated circuits.

In the discussion that follows, a photo-mask should be understood toinclude: a chromium-on-glass photo-mask, an alternating phase-shiftingphoto-mask, an attenuating phase-shifting photo-mask, and/ormultiple-exposure photo-masks (e.g., where patterns printed on a waferor semiconductor die using two or more photo-masks are combined toproduce a desired or target pattern, such as a portion of an integratedcircuit). Furthermore, a mask pattern should be understood to includethe pattern of spatially varying transmittance magnitude and/ortransmittance phase in a given photo-mask. Note that, typically, themanufactured or fabricated mask pattern in a given photo-mask deviatesfrom an ideal target mask pattern, for example, because of defects thatcan occur during the photo-mask fabrication process.

In addition, in the discussion that follows note that an image and/or apattern may include a bitmap or grayscale file that includes a set ofvalues corresponding to pixels in the image and/or the pattern.Furthermore, the quantization (i.e., the number of bits) in these filesmay be varied, as needed, during the measurements and calculations thatare described. Alternative formats having the same or similarinformation content, including a vector-based format such as a GraphicDesign System II (GDSII) and/or an OASIS format, may be used in someembodiments of the images and/or patterns. And in some embodiments, theimages and/or patterns include real and imaginary components (orequivalently, magnitude and phase information).

We now describe embodiments of the calculation technique. FIG. 1Apresents a flow chart illustrating a method 100 for determining areconstructed mask pattern, which is performed by a computer system(such as computer system 700 in FIG. 7). During operation, the computersystem receives a target mask pattern (operation 110) and an image of atleast a portion of a photo-mask corresponding to a mask pattern(operation 112), where the image is determined using inspection opticsthat includes an optical path. For example, the image may be aphoto-mask inspection image that is obtained using an optical inspectionsystem, such as a photo-mask inspection tool (e.g., the TeraScan™photo-mask inspection system from KLA-Tencor, Inc., of San Jose,Calif.).

Then, the computer system reconstructs the mask pattern from the imagebased on a characteristic of the optical path and the target maskpattern (operation 114). For example, the characteristic may include arange of wavelengths transmitted via the optical path (such as anoptical bandwidth of the optical path). Note that the mask pattern maybe reconstructed using a constrained inverse optical calculation inwhich there are a finite number of discrete feature widths allowed inthe reconstructed mask pattern, and where a given feature has a constantfeature width. As described further below with reference to FIG. 4, inthe constrained inverse optical calculation the image may be at an imageplane of a model of the optical path and the mask pattern may be at anobject plane of the model of the optical path. Furthermore, note thatthe reconstructed mask pattern may be characterized by additionalspatial frequencies than the image. Thus, the reconstructed mask patternmay recover information in the mask pattern that was lost or distorted,e.g., by correcting image artifacts, when the image was measured, forexample, because of a finite numerical aperture in the opticalinspection system. For example, spatial frequencies in the image may beapproximately greater than or equal to the average critical dimension ofthe target mask pattern. In contrast, spatial frequencies in thereconstructed mask pattern may correspond to a resolution that is 1% (orless) of an average critical dimension of the target mask pattern.

As shown in FIG. 2A, which illustrates a reconstructed mask pattern 200,in some embodiments features 210 in reconstructed mask pattern 200 mayeach have constant feature width 212-1 (as opposed to a variable width,such as those associated with the edge steps illustrated by dashed line214). For example, constant feature width 212-1 may be an averagecritical dimension of reconstructed mask pattern 200. Note that theaverage critical dimension may be associated with a minimum of a costfunction in the inverse optical calculation, where the cost functioncorresponds to a difference between the reconstructed mask pattern (suchas a simulated image of the reconstructed mask pattern) and the actualmask pattern (such as an image of the mask pattern). In this example,the finite number of discrete feature widths in reconstructed maskpattern 200 includes one feature width: feature width 212-1.

Alternatively, as shown in FIG. 2B, which illustrates a reconstructedmask pattern 250, primary features (such as primary feature 260) inreconstructed mask pattern 250 may each have constant feature width212-1, and sub-resolution assist features or SRAFs (such as SRAF 262) inreconstructed mask pattern 250 may each have another constant featurewidth 212-2, which is different than constant feature width 212-1. Inthis example, the finite number of discrete feature widths inreconstructed mask pattern 250 includes two feature widths: featurewidth 212-1 and feature width 212-2.

In this way, the reconstructed mask pattern obtained via the inverseoptical calculation may constitute an efficient simplification of theactual mask pattern, and in the process may provide information (such asthe average critical dimension) that can be used to qualify thephoto-mask. In particular, referring back to FIG. 1A, in someembodiments the computer system optionally compares the average criticaldimension of the reconstructed mask pattern to the critical dimension ofthe target mask pattern to determine a critical-dimension bias of thephoto-mask (operation 116), which serves as an indicator ofcritical-dimension uniformity. Furthermore, the computer system mayoptionally determine an acceptance condition of the photo-mask based onthe critical-dimension bias (operation 118). For example, if thecritical-dimension bias exceeds an absolute or a relative criteria (suchas 10 nm or 5% of the critical dimension), the photo-mask may berejected, subject to rework, or subject to additional qualificationtesting.

As discussed further below with reference to FIG. 5, in some embodimentsthe computer system uses the reconstructed mask pattern to determine oneor more aerial image(s) on a wafer using a forward optical calculation.In particular, the computer system may calculate the aerial image(s) atan image plane of a model of an optical path (such as the optical pathassociated with an exposure tool in a photolithographic process) usingthe reconstructed mask pattern and appropriate illumination by a sourcepattern (annular, quadrapole, arbitrary, etc.), at an object plane ofthe model of the optical path associated with the exposure tool. Thissimulation may use conditions associated with the photolithographicprocess (such as immersion optics, a disk illumination with a sigma of0.75, a numerical aperture of 1.35, a wavelength of 193 nm, etc.).Furthermore, the computer system may calculate an estimated resistpattern on the wafer based on the determined aerial image(s) using amodel of a photoresist in the photolithographic process, such as a modelof a positive or a negative resist. Then, the computer system mayidentify differences between the estimated resist pattern and a targetpattern (such as a portion of a circuit that is to be printed on a waferor a semiconductor die), and may determine the acceptance condition ofthe photo-mask based on the identified differences. However, in someembodiments where the estimated resist patter is not calculated, thedifferences may be between the aerial image(s) and the target pattern.

Note that in addition to efficiently determining the critical-dimensionbias of the photo-mask, by constraining the inverse optical calculationin method 100, a convergence time of the inverse optical calculation maybe faster than that of an unconstrained inverse optical calculation. Inaddition, the average critical dimension determined when reconstructingthe mask pattern in the constrained inverse optical calculation may beless noisy and more reproducible than the results obtained when the maskpattern is reconstructed in an unconstrained inverse optical calculation(which has a much larger number of degrees of freedom, such as thepossible feature geometries or positions of edges of the features). Forexample, the accuracy of the average critical dimension determined usingthe constrained inverse optical calculation may have a standarddeviation less than 0.5 nm in the reconstructed mask pattern, as opposedto a standard deviation of 0.8 nm in the reconstructed mask pattern whenthe unconstrained inverse optical calculation is used.

A variation on the calculation technique is shown in FIG. 1B, whichpresents a flow chart illustrating a method 150 for determining areconstructed mask pattern, which is performed by a computer system(such as computer system 700 in FIG. 7). During operation, the computersystem receives the target mask pattern (operation 110) and receives animage of at least the portion of the photo-mask corresponding to themask pattern (operation 160), where spatial frequencies in the image areapproximately greater than or equal to the average critical dimension ofthe target mask pattern. Then, the computer system reconstructs the maskpattern from the image based on a characteristic of the optical path andthe target mask pattern (operation 162) using the constrained inverseoptical calculation in which there are the finite number of discretefeature widths allowed in the reconstructed mask pattern, and where thegiven feature has the constant feature width. Next, the computer systemdetermines the critical-dimension bias of the photo-mask from the finitenumber of discrete feature widths (operation 164).

In some embodiments of methods 100 (FIG. 1A) and 150 there may beadditional or fewer operations. Moreover, the order of the operationsmay be changed, and/or two or more operations may be combined into asingle operation. For example, instead of the image, in some embodimentsthe constrained inverse optical calculation uses a set of images. Theseimages may be determined at different focal conditions in the opticalinspection system (such as at different image planes or surfaces). Moregenerally, the set of images may be obtained using different imagingconditions, such as: different wavelengths, different focal conditions,different illumination types (such as annular, quadrapole, immersion,etc.), different measurement techniques, etc.

FIG. 3 presents a block diagram illustrating a photo-mask inspection andqualification system 300. In this system, one or more inspection imagesof the photo-mask are obtained using optical inspection system 310.

Then, an analysis system 312 determines the reconstructed mask patternbased on the one or more inspection images and a target mask pattern. Asnoted previously, and described further below with reference to FIG. 4,in some embodiments the reconstructed mask pattern is determined usingthe constrained inverse optical calculation based on information aboutan optical path associated with optical inspection system 310, includingthe characteristic of this optical path. In the process, analysis system312 may recover lost information in the one or more inspection images(for example, due to a finite numerical aperture in optical inspectionsystem 310) and/or to correct for image artifacts, such as those due tofocal errors and/or optical aberrations. Therefore, the reconstructedmask pattern may include additional spatial frequencies than the one ormore inspection images.

Additionally, photo-mask qualifier 316 may use the constant featurewidth(s) in the reconstructed mask pattern to determine acritical-dimension bias of the photo-mask, and may determine anacceptance condition of the photo-mask based on the critical-dimensionbias. For example, the photo-mask may be acceptable if thecritical-dimension bias is less than 10 nm or 5% of a critical dimensionin the target mask pattern. Alternatively, if the photo-mask is notacceptable, it may be rejected, subject to rework, or subject toadditional qualification testing.

In some embodiments, an optional lithography simulator 314 calculatesone or more aerial images based on the reconstructed mask pattern and asource pattern that ‘illuminates’ the reconstructed mask pattern duringthe calculation. (Alternatively, optional lithography simulator 314 maycalculate the one or more aerial images based on the source pattern, thetarget mask pattern and a mask-defect pattern, which was determined bycomparing the target mask pattern and the reconstructed mask pattern.)As noted previously, and described further below with reference to FIG.5, in some embodiments the one or more aerial images are calculatedusing a forward optical calculation based on the information about anoptical path associated with the photolithographic process (such as amodel of the optical path in the exposure tool). Furthermore, in someembodiments optional lithography simulator 314 may calculate anestimated resist pattern based on the one or more aerial images and themodel of the photoresist in the photolithographic process. Thus, in someembodiments optional lithography simulator 314 calculates a simulated orestimated wafer pattern that can be produced in the photolithographicprocess.

Then, photo-mask qualifier 316 may identify a difference(s) between theone or more aerial images and/or the estimated resist pattern and thetarget pattern, and may determine the acceptance condition of thephoto-mask based on the identified difference(s). For example,photo-mask qualifier 316 may analyze the one or more aerial imagesand/or the estimated resist pattern to determine if it is acceptable,e.g., if differences with respect to the target pattern and/or anydefects that are present are within acceptable bounds, such as afraction of a critical dimension in the target pattern. (In embodimentswhere the one or more aerial images are used, this may involve priorcorrelation with the critical dimension(s) of a test wafer.) If yes, thephoto-mask may be accepted, and if not the photo-mask may be rejected,subject to rework, or subject to additional qualification testing. Notethat, in some embodiments, photo-mask acceptance is also based, at leastin part, on a comparison of the reconstructed mask pattern and at leasta portion of the target mask pattern for the same or a differentphoto-mask (which is sometimes referred to as ‘die-to-databaseinspection’).

In some embodiments, the photo-mask is accepted (or not) based on aprocess window (such as a range of exposure times, a depth of focus, arange of exposure intensities, and/or a normalized image log slope)associated with the reconstructed mask pattern. In this way, aparticular defect in the photo-mask that is fatal when slightlyoverexposed may be identified, even though it is negligible at thenominal dose. In addition, in some embodiments the photo-mask isaccepted (or not) based on features in the one or more aerial imagesand/or the estimated resist pattern that are identified over orthroughout the process window and/or based on an impact on a criticaldimension across the process window. Note that acceptance of thephoto-mask may be fully automated, while in other embodiments it may notbe fully automated. Thus, information (such as identifiedcritical-dimension bias, differences and/or features) may be presentedto an operator, who may verify an acceptance condition made byphoto-mask inspection and qualification system 300 (FIG. 3) or who mayindependently determine whether or not to accept the photo-mask.

Alternatively or additionally, the photo-mask may be qualified based oncomparisons between the estimated resist pattern and actual patternedwafer patterns produced or generated using the photo-mask. For example,an optional wafer-exposure system 318 (i.e., the exposure tool) may beused to produce a printed wafer using the photo-mask, and a printedwafer image of the pattern on the printed wafer may be determined usingan optional wafer-imaging system 320 (such as the PUMA™ patternedwafer-inspection platform from KLA-Tencor, Inc., of San Jose, Calif.).However, this brute-force approach is often expensive and timeconsuming. In addition, errors introduced in the photolithographicprocess in optional wafer-exposure system 318 may reduce the accuracy ofthe qualification decision made by photo-mask qualifier 316.

Thus, the calculation technique may overcome the limitations of CD-SEMimages by providing a high-throughput and efficient technique withoutrequiring the use of additional equipment (i.e., the mask inspectionsystem can be used to perform mask inspection) for determining whetheror not the photo-mask (including any defects) is acceptable.Consequently, the calculation technique may improve photo-mask and/orwafer yield, and thus may decrease the cost and time to market ofphoto-masks and integrated circuits.

FIG. 4 presents a block diagram illustrating an inverse opticalcalculation 400. In this calculation, a predicted input 410 (such as apredicted reconstructed mask pattern) at an object plane of optical path412 is determined based on an output 414 (such as an inspection image)at an image plane of optical path 412. For example, the reconstructedmask pattern at the object plane may be determined from an inspectionimage using the target mask pattern and information about the opticalpath in optical inspection system 310 (FIG. 3). Note that informationabout optical path 412 may include the characteristic of optical path412, such as an optical bandwidth of optical path 412.

In particular, the reconstructed mask pattern of the photo-mask at theobject plane may be determined from the inspection image at the imageplane and the target mask pattern at object plane, as well as frominformation about the optical path in optical inspection system 310(FIG. 3). In particular,R=I ⁻¹ IM,where I is a forward optical path (described with reference to FIG. 5below), I⁻¹ is an inverse optical path operator, M is the actual(physical) mask pattern, and the application of I to M is the inspectionimage, and R is the reconstructed mask pattern. As noted previously, thereconstructed mask pattern may be characterized by additional spatialfrequencies than the inspection image (e.g., it includes additionalmagnitude and/or phase information at one or more additional spatialfrequencies). For example, if the inspection image may be characterizedby a band of spatial frequencies, the reconstructed mask pattern may becharacterized by another band of spatial frequencies that includes theband of spatial frequencies, and which approximately equals the band ofspatial frequencies associated with the photo-mask.

While the preceding discussion illustrates the inverse opticalcalculation using a single output 414, in other embodiments two or moreinspection images at image plane(s) of optical path 412 may be used. Forexample, instead of the inspection image, there may be a set ofinspection images that are each determined using different wavelengths,different focal conditions (e.g., on different focal surfaces orplanes), and/or different imaging conditions in optical inspectionsystem 310 (FIG. 3). These inspection images may include magnitudeand/or phase information. For example, inspection images that includemagnitude and relative phase information may be measured by generatingan interference pattern using measurement and reference beams derivedfrom a common light source or that are spatially and temporallycoherent. Alternatively, phase contrast optics may be utilized. In someembodiments, the difference of two inspection images at image plane(s)may be used in the inverse optical calculation 400. Furthermore, in someembodiments each of the inspection images at the image plane(s) used inthe inverse optical calculation 400 or a term(s) including somecombination of the inspection images at the image plane(s) may bemultiplied by a corresponding weight. In this way, inverse opticalcalculation 400 may emphasize one or more of the inspection images at animage plane relative to other inspection images (at the same or otherimage planes) used in inverse optical calculation 400.

In some embodiments, inverse optical calculation 400 is based oniterative minimization of an error function (H), which is also sometimesreferred to as a ‘cost function’ or a ‘Hamiltonian function.’ Inparticular, during each iteration of inverse optical calculation 400 theerror function may be a function of the difference between output 414and a pattern (or image) that results when input 410 is projectedthrough optical path 412. In some embodiments, input 410 initiallycorresponds to the target mask pattern, and as the calculationprogresses this pattern is allowed to evolve while output 414 is heldconstant (subject to the constraint that there are a finite number ofdiscrete feature widths allowed in input 410, and where a given featurehas a constant feature width). In embodiments with multiple patterns (orimages) at object plane(s) and/or image plane(s), the error function (H)equals

${\sum\limits_{j = 1}^{N}{w_{j}{{I_{j} - I_{oj}}}^{n}}},$where I_(j) is the forward projection of the jth reconstructed maskpattern at the object plane (out of N patterns in this example) throughoptical path 412, w_(j) is a corresponding weight, I_(oj) is the jthinspection image at an image plane, and n is a power. Note that theerror function (II) approaches zero as I_(j) approaches I_(oj).

In an exemplary embodiment, N is 3 and n is 2. Three patterns (orinspection images) at the image plane(s) may be determined at threedifferent focal conditions (or focus settings) in optical inspectionsystem 310 (FIG. 3). For example, with a wavelength of 260 nm, the focalconditions may be at −600 nm (relative to nominal focus), at 0 nm (i.e.,at nominal focus), and 600 nm (relative to nominal focus). Alternativelyor in addition, the three patterns (or inspection images) at the imageplane(s) may be determined at three different wavelengths or imagingconditions. Furthermore, a corresponding set of weights {w_(j)} may be1, 0.1, and 1.

In other embodiments, the weights are varied as inverse opticalcalculation 400 progresses and/or different weights are used forspecific parts (or even pixels) of a pattern. For example, the weightsmay be determined based on the difference between I_(j) and I_(oj) at agiven step in inverse optical calculation 400. This approach mayexaggerate the features or defects, especially when inverse opticalcalculation 400 is close to a local or global minimum and the errorfunction (H) corresponds to small differences. Thus, in general theerror function (H) may be expressed as a double integral over thepattern or image area and there may be separate time-dependent weightsfor I_(j) and I_(oj). Furthermore, in some embodiments the errorfunction (H) is expressed as a relative difference between I_(j) andI_(oj) for at least a portion of inverse optical calculation 400 as itprogresses.

It will be recognized by one of ordinary skill in the art that inverseoptical calculation 400 described above is poorly defined. Inparticular, numerous possible reconstructed mask patterns at the objectplane may result from the same observed output 414. Therefore, input 410may be selected such that it is ‘most likely’ to represent the actualphoto-mask. A variety of constraints and additional criteria may beimposed when determining the solution(s) to this problem in order tofind a unique answer(s). For example, input 410 may be that which hasthe smallest value of the error function (H).

Other constraints based on a priori knowledge of the photo-maskmanufacturing process may also be applied to resolve the ambiguity amongseveral competing candidate defect possibilities (i.e., differentpossible solutions for input 410). For example, there may be a prioriknowledge about typical defect types (including the distribution ofdefect sizes and phases) that arise during the photo-mask manufacturingprocess. In addition, information may also be obtained from neighboringdefects on the photo-mask that is being inspected, or from previousphoto-masks that were manufactured on the same process line andinspected. For example, given the likelihood that point defects tend tobe generated by common mechanisms, a common link between more than onesuch defects may constrain the possible solution options in inverseoptical calculation 400.

One common type of defect is known as a critical-dimension defect or asizing error. This type of defect is not an isolated feature (i.e., afeature where one does not belong), or a missing feature (i.e., whereone was expected), but rather an error in the dimension of the featurebeing patterned on the photo-mask. In addition, the large mask errorenhancement factors (MEEFs) of leading-edge lithographic processes makesit important to understand how such observed critical-dimension defectson or in photo-masks impact wafer manufacturing (i.e., the printedpatterns on wafers). By determining the average critical dimension whilecalculating the reconstructed mask pattern, the calculation technique iswell suited to identifying and assessing the impact (i.e., thesignificance) of these and other defects in the photo-masks.

FIG. 5 presents a block diagram illustrating a forward opticalcalculation 500. In this calculation, a suitable illuminated input 510(such as the reconstructed mask pattern) at an object plane of opticalpath 512 is used to determine a predicted output 514 (such as an aerialimage) at an image plane of optical path 512. For example, using thereconstructed mask pattern, a source pattern, and information about anoptical path associated with optional wafer-exposure system 318 (FIG.3), the aerial image can be determined. Note that optical path 512 maybe different than optical path 412 (FIG. 4). In general, informationabout optical path 512 may include some or all of the aspects of thephotolithographic process, such as illumination settings, theelectromagnetics of the photo-mask, the exposure-tool optics, opticaleffects, etc. In addition, in some embodiments forward opticalcalculation 500 models the effect of a photoresist, including flareand/or etch effects.

Note that calculations corresponding to one or more optical paths ininverse optical calculation 400 (FIG. 4) and/or forward opticalcalculation 500 may be implemented using Fourier-optical techniques.Furthermore, the optical paths in inverse optical calculation 400 (FIG.4) and/or forward optical calculation 500 may include multiple models ofoptical paths (such as when the set of second inspection images includesinspection images from two or more different optical inspectionsystems), which are then used to determine the reconstructed maskpattern. Also note that while optical path 412 (FIG. 4) and optical path512 have been traversed in particular directions, each of these opticalpaths may be traversed in either direction.

We now describe an exemplary embodiment of the forward opticalcalculation or forward projection operation used to calculate the aerialimage and/or the estimated resist pattern. For simplicity, coherentillumination of the estimated photo-mask is utilized. Furthermore, theelectric field falling upon the photo-mask is approximately constant.Thus, the clear regions of the photo-mask pass the light, while theopaque regions block the light. It follows that a scalar electric fieldE, just behind the photo-mask, may be expressed as

${{E\left( \overset{->}{r} \right)} = \begin{Bmatrix}0 & {chrome} \\1 & {glass}\end{Bmatrix}},$where {right arrow over (r)}=(x, y) is a point on the (x, y) plane. Asdiscussed below with reference to FIG. 6, this representation of thephoto-mask may be re-expressed using a function φ (referred to as alevel-set function) having positive regions that indicate glass andnegative regions that indicate chrome. Furthermore, the level-setfunction may equal zero at the boundaries or contours of the photo-mask.Therefore, the electric field E associated with the photo-mask may bere-expressed as a function of this level-set function, i.e.,E({right arrow over (r)})=ĥ(φ(x,y)),where ĥ is the Heaviside function

${\hat{h}(x)} = {\begin{Bmatrix}1 & {x \geq 0} \\0 & {x < 0}\end{Bmatrix}.}$

Since an ideal diffraction limited lens acts as a low-pass filter, thismay be used as an approximation to the actual (almost but not quiteperfect) lens in the optical path of optional wafer-exposure system 318in FIG. 3 (in this example). Mathematically, the action of the lens maybe expressed asA({right arrow over (r)})=f⁻¹(Ĉ(f(E({right arrow over (r)}))))where A({right arrow over (r)}) indicates the electric fielddistribution on the wafer, f indicates the Fourier transform, f⁻¹indicates the inverse Fourier transform, and Ĉ indicates the pupilcutoff function, which is zero for frequencies larger than a thresholddetermined by the numerical aperture of the lens, and one otherwise.Thus, the pupil function is

${{\overset{\Cap}{C}\left( {k_{x},k_{y}} \right)} = {{\hat{h}\left( {k_{\max}^{2} - \left\lbrack {k_{x}^{2} + k_{y}^{2}} \right\rbrack} \right)} = \begin{Bmatrix}0 & {{k_{x}^{2} + k_{y}^{2}} \geq k_{\max}^{2}} \\1 & {{k_{x}^{2} + k_{y}^{2}} < k_{\max}^{2}}\end{Bmatrix}}},$wherein k_(x), k_(y) and k_(max) represent frequency coordinates inFourier space. Therefore, the aerial image (at the wafer in optionalwafer-exposure system 318 in FIG. 3) is simply the square of theelectric fieldI({right arrow over (r)})=|A({right arrow over (r)})|².

Combining these two equations, we findF(φ(x,y))=(|f⁻¹(Ĉ(f(ĥ(φ(x,y)))))|²).This is a self-contained formula for the aerial image obtained byoptional wafer-exposure system 318 (FIG. 3).

Note that this is just one embodiment of the forward projector that canbe used within the scope of this disclosure, chosen by way of exampledue to its relative simplicity. More sophisticated forward models alsofall within the scope of the present disclosure. Such models may takeinto account, by way of example but not limitation, various illuminationconditions (e.g., off-axis, incoherent), the actual electromagnetics ofthe light field interacting with the photo-mask, various types ofphoto-masks other than chrome on glass (e.g., attenuated phase shifting,strong phase shifting, other materials, etc.), the polarization of thelight field, the actual properties of the lens (such as aberrations),and/or the vector nature of the electromagnetic field as it propagatesthrough the optical path.

We now describe the level-set functions in more detail. As notedpreviously, the reconstructed mask pattern being determined in theconstrained inverse optical calculation 400 may be represented as afunction having a set of values that is larger than those in theinspection image. In some embodiments, the function is a level-setfunction. This is illustrated in FIG. 6, which provides a mask pattern600 and corresponding level-set functions 614. This mask patternincludes alternating regions with glass (610-1) and chromium (610-2).Transitions from one region to another are characterized by a contour oran edge, such as edge 612. When viewed from a direction perpendicular toa plane of the photo-mask, edge 612 defines mask pattern 600.

Level-set function 614-1 has two values 616. Furthermore, edge 612 maycorrespond to a mid-point between these two values 616. In contrast,level-set function 614-2 has three values 618, and edge 612 maycorrespond to value 618-2. While not illustrated in FIG. 6, level-setfunctions 614 extend into the plane of FIG. 6 (e.g., they may be3-dimension functions). As is known to one of skill in the art, thereare many alternate level-set functions and/or configurations that may beused. For example, in some embodiments one or more separate level-setfunctions and/or separate patterns or images may be used for thefeatures or defects.

As illustrated by level-set function 614-2, in some embodiments thelevel-set function may be expressed as a signed distance functionrelative to the contour or edge 612 (e.g., the value of the level-setfunction in at least a region is a function of the distance from theedge 612). This formulation may allow effects that occur nearer to theedge 612 (such as critical-dimension defects) to be highlighted.However, because features or defects in photo-masks may occur at randomlocations (including those far removed from edge 612), level-setfunction 614-1 may be useful in that it provides an equal weighting withrespect to edge 612.

In some embodiments, during each iteration of inverse opticalcalculation 400 (FIG. 4) the level-set function corresponding to input410 (FIG. 4) being modified is updated according toφ_(i+1)−φ_(i) +Δt·∇(H),where φ_(i+1) is an updated version of the level-set function, φ_(i) isthe current version of the level-set function, Δt is a step size in thecalculation and ∇(H) is a gradient or a derivative of the errorfunction. In an exemplary embodiment, ∇(H) is

${\frac{\delta\; H}{\delta\;\phi}}_{\varphi_{i}},$i.e., it is the Frechet derivative of the error function H. Furthermore,in some embodiments ∇(H) is the direction of steepest descent forminimizing or optimizing H by changing φ. Furthermore, in someembodiments a 1^(st) order and/or a 3^(rd) order Runge-Kutta method isused when updating φ_(i). In other embodiments, a Conjugate Gradienttechnique, a Levenberg-Marquardt technique, a Quasi-Newton technique,and/or a Simplex technique may be used.

At least some aspects of Simulated Annealing may be utilized in someembodiments of inverse optical calculation 400 (FIG. 4). In particular,the error function H may be allowed to increase during some steps as thecalculation evolves. In this way, the global minimum in themulti-dimensional space may be determined. Note that the size of thismulti-dimensional space may be a number of quantization levels to thepower of the number of pixels in the reconstructed mask pattern or inthe inspection image. In an exemplary embodiment, the pattern or imagehas at least 1 million pixels (for example, 1024×1024).

In one embodiment, in any iteration of inverse optical calculation 400(FIG. 4), changes in φ that decrease or increase the error function (H)up to 0.5% are performed. If a larger change will result (e.g.,ΔH>0.5%), the step size Δt may be decreased by a factor that is at leastgreater than 1 and the change in φ is implemented (or not) based on aprobability and a value P given by

${\mathbb{e}}^{\frac{- {kH}_{i + 1}}{H_{i}}},$where H_(i+1) is the error function in the i+1^(th) iteration (if thechange in φ is implemented) and H_(i) is the error function in i^(th)iteration (note that the ratio of H_(i+1)/H_(i) equals 1+ΔH). In someembodiments k is 0.155. For example, if the value P is 0.3 and theprobability is a random (or pseudorandom) number between 0 and 1 that isless than P, the error function may be increased before proceeding. Inthis way, inverse optical calculation 400 (FIG. 4) initially takes largesteps and thereby explores the solution space.

Furthermore, in some embodiments inverse optical calculation 400 (FIG.4) is divided into a series of overlapping sub-problems (also referredto as ‘work units’) at least some of which are processed independentlyand/or concurrently. These work units may be based on elements orstructures (for example, repetitive structures) in the reconstructedmask pattern, the target pattern, and/or in the inspection image. Insome embodiments, the works units are selected such that there is aprobability exceeding a pre-defined value (e.g., a high probability)that most if not all of the work units include at most one defect (forexample, the work units may be based on differences between aninspection image and a simulated inspection image that is determinedusing the target mask pattern). Furthermore, in some embodiments thework units may partially overlap neighboring work units. For example,the work units may be between 10,000 nm² and 100 μm² in size.

In some embodiments, inverse optical calculation 400 (FIG. 4) is run for100, 1000 or 10,000 iterations at which point the optimal solution hasbeen determined. In other embodiments, the calculation is stopped basedon convergence criteria, such as oscillatory behavior, a relative and/orabsolute difference between the inspection image and images that resultwhen the reconstructed mask pattern is projected through optical path412 (FIG. 4), the latest change to the error function H, and/or thehistory of changes to the error function H. For example, the relativedifference may be less than 1% and/or the absolute difference may be 10nm for a critical dimension of 100 nm. Note that in some embodiments,the level-set function is re-distanced (i.e., restored to one having thedistance function property relative to the edge 612) at intermediateiterations during inverse optical calculation 400 (FIG. 4). In anexemplary embodiment, such re-distancing occurs at least every 20iterations (for example, every 14 iterations).

Using this inverse calculation approach, features smaller than thewavelength of the light source used to perform optical measurements orto print wafer patterns in the photolithographic process may bedetermined. For example, in simulations using a light source having awavelength of 260 nm, features and defects as small as (40 nm)² on aphoto-mask were determined.

We now discuss computer systems for implementing the calculationtechnique. FIG. 7 presents a block diagram illustrating a computersystem 700. Computer system 700 includes one or more processors 710, acommunication interface 712, a user interface 714, and one or moresignal lines 722 coupling these components together. Note that the oneor more processors 710 may support parallel processing and/ormulti-threaded operation, the communication interface 712 may have apersistent communication connection, and the one or more signal lines722 may constitute a communication bus. Moreover, the user interface 714may include a display 716, a keyboard 718, and/or a pointer 720, such asa mouse.

Memory 724 in the computer system 700 may include volatile memory and/ornon-volatile memory. More specifically, memory 724 may include ROM, RAM,EPROM, EEPROM, flash, one or more smart cards, one or more magnetic discstorage devices, and/or one or more optical storage devices. Memory 724may store an operating system 726 that includes procedures (or a set ofinstructions) for handling various basic system services for performinghardware dependent tasks. The memory 724 may also store procedures (or aset of instructions) in a communication module 728. The communicationprocedures may be used for communicating with one or more computersand/or servers, including computers and/or servers that are remotelylocated with respect to the computer system 700.

Memory 724 may also include multiple program modules (or a set ofinstructions), including: analysis module 730 (or a set ofinstructions), lithography simulator 732 (or a set of instructions),and/or photo-mask qualifier 734 (or a set of instructions). Note thatone or more of these program modules (or sets of instructions) mayconstitute a computer-program mechanism. Furthermore, note that one ormore of these program modules (or sets of instructions) may beimplemented as a stand-alone software application, or as a programmodule or subroutine in another application, such as photo-maskinspection software.

During operation, computer system 700 may receive inspection image(s)736 associated with a photo-mask. FIG. 8 presents a block diagramillustrating an image data structure 800. This image data structure mayinclude information corresponding to one or more images 810 (such asinspection image(s) 736). For a given image, such as image 810-1, imagedata structure 800 may include: a focal plane or focus condition 812-1at which image 810-1 was acquired, other imaging conditions 814-1 atwhich the image 810-1 was measured, and spatial variation in magnitude816-1 and/or phase 818-1 information in image 810-1.

Referring back to FIG. 7, analysis module 730 may determinereconstructed mask pattern 738 in a constrained inverse opticalcalculation using inspection image(s) 736, target mask pattern 740,information about a first optical path 742 associated with opticalinspection system 310 (FIG. 3), and one or more discrete feature widths744. Furthermore, by comparing a determined average critical dimension746 of reconstructed mask pattern 738 with a target average criticaldimension 748 of target mask pattern 740, analysis module 730 maydetermine critical-dimension bias 750. In some embodiments, by comparingreconstructed mask pattern 738 and target mask pattern 740, analysismodule 730 may determine defect(s) 752 in reconstructed mask pattern738.

Furthermore, in some embodiments lithography simulator 732 may calculateone or more aerial image(s) 754 in a forward optical calculation usingreconstructed mask pattern 738, information about a second optical path756 (such as that associated with optional wafer-exposure system 318 inFIG. 3), and photolithographic conditions 758 (including a sourcepattern). Furthermore, lithography simulator 732 may calculate estimatedresist pattern(s) 760 using the one or more aerial image(s) 754 and aphotoresist model 762. After at least some of these calculations areperformed, photo-mask qualifier 734 may identify a difference(s) 764 (orfeatures) between the one or more aerial image(s) 754 and/or estimatedresist pattern(s) 760 and target pattern(s) 766 (such as portions of acircuit).

Using critical-dimension bias 750, defect(s) 752 and/or difference(s)764, photo-mask qualifier 734 may determine an acceptance condition 768of the photo-mask.

Instructions in the various modules in memory 724 may be implemented ina high-level procedural language, an object-oriented programminglanguage, and/or in an assembly or machine language. The programminglanguage may be compiled or interpreted, i.e, configurable or configuredto be executed, by the one or more processors 710.

In some embodiments, at least some of the information in memory 724 isencrypted. For example, the lithographic simulator 732 and/or its outputfiles (such as estimated resist pattern(s) 760) may be encrypted so thatintegrated-circuit manufacturers are more willing to share thisinformation with photo-mask shops (where photo-mask inspection may beperformed). As discussed further below, in an alternate approach, thephoto-mask shop may send the photo-mask images (e.g., inspectionimage(s) 736) to integrated-circuit manufacturers, who may performwafer-pattern simulations and/or may determine photo-mask acceptance.Therefore, information ‘stored’ in memory 724 in FIG. 7 may be storedlocally and/or at remote locations.

Although the computer system 700 is illustrated as having a number ofdiscrete items, FIG. 7 is intended to be a functional description of thevarious features that may be present in the computer system 700 ratherthan as a structural schematic of the embodiments described herein. Inpractice, and as recognized by those of ordinary skill in the art, thefunctions of the computer system 700 may be distributed over a largenumber of servers or computers, with various groups of the servers orcomputers performing particular subsets of the functions. In someembodiments, some or all of the functionality of the computer system 700may be implemented in one or more ASICs, one or more field programmablegate arrays (FPGAs), and/or one or more digital signal processors(DSPs). In some embodiments the functionality of the computer system 700may be implemented more in hardware and less in software, or less inhardware and more in software, as is known in the art.

In the preceding discussion, a ‘computer system’ may include a varietyof devices, such as: a personal computer, a laptop computer, a mainframecomputer, a portable electronic device, a server and/or a clientcomputer (in a client-server architecture), and/or other device capableof manipulating computer-readable data or communicating such databetween two or more computing systems over a network (such as theInternet, an Intranet, a LAN, a WAN, a MAN, or combination of networks,or other technology enabling communication between computing systems).

In some embodiments, reconstructed mask pattern 200 (FIG. 2A),reconstructed mask pattern 250 (FIG. 2B), photo-mask inspection andqualification system 300 (FIG. 3), inverse optical calculation 400 (FIG.4), forward optical calculation 500 (FIG. 5), mask pattern 600 (FIG. 6)and corresponding level-set functions 614 (FIG. 6), computer system 700,and/or image data structure 800 (FIG. 8) include fewer or additionalcomponents. Furthermore, in these embodiments two or more components arecombined into a single component and/or a position of one or morecomponents may be changed.

As discussed previously, at least a portion of the calculation techniquedescribed may be implemented at a remote location. For example, theinspection image may be measured at a first location, and then providedto computer system 700 at a second (remote) location. After calculatingthe reconstructed mask pattern, the aerial image and/or the estimatedresist pattern, and determining a critical-dimension bias, identifyingdifferences or features and/or determining an acceptance condition ofthe photo-mask, results may be reported back to the first location. Thisapproach may allow photo-mask shops and integrated-circuit manufacturersto work together to determine whether or not a photo-mask is acceptable,or should be reworked or rejected. Historically, both of these partieshave had reservations about such an arrangement. Photo-mask shops may bereluctant because it places the ability to reject a photo-mask in thehands of the end user (the integrated-circuit manufacturer), who may becautious about accepting a photo-mask and may not have a financialmotivation to accept a photo-mask that is less than optimal. Inparticular, since there is no cost to the end user, any potential defectmay result in a photo-mask being rejected and the photo-mask shop may beforced to rewrite the photo-mask at their expense.

The approach described above may help resolve this conflict by creatinga computational infrastructure that is agreed upon by both thephoto-mask shop and the integrated-circuit manufacturer. In embodimentswhere photo-mask acceptance is fully automated, computer system 700 maybe installed at the integrated-circuit manufacturer and inspectionimages sent by the photo-mask shop may be processed without exposing thedetails of the integrated-circuit manufacturing process to thephoto-mask maker, and yet at the same time without exposing thephoto-mask shop to the human judgment of the integrated-circuitmanufacturer in accepting or rejecting photo-masks.

Alternatively, the photo-mask shop may calculate the reconstructed maskpattern from the inspection image, which is then sent to theintegrated-circuit manufacturer where simulations of wafer patternsunder various process conditions may be performed. The results may besent back to the photo-mask shop, where they may be used to determinethe disposition of defects (e.g., whether or not the photo-mask isacceptable). Therefore, the embodiments of the system and methoddescribed herein may be implemented by the photo-mask shop and/or by theintegrated-circuit manufacturer.

While the preceding discussion used a photo-mask as an illustrativeexample, in other embodiments the calculation technique is applied toother types of images. For example, the calculation technique may beapplied during the inspection and qualification of a patterned wafer (orsemiconductor die) that is fabricated using a photolithographic ordirect write process. In particular, a reconstructed wafer inspectionimage of the patterned wafer may be determined from a wafer inspectionimage (which may be obtained using optional wafer imaging system 320 inFIG. 3) and at least a portion of the target pattern. More generally,the calculation technique may be used in a wide variety of imagingand/or measurement applications that are based on wave phenomenapropagating in different types of media (such as electromagnetic wavesand sound waves) and at different ranges of wavelengths (such as audio,radio, microwave, infrared, visible, ultra violet, and x-ray).

The foregoing descriptions of embodiments of the present invention havebeen presented for purposes of illustration and description only. Theyare not intended to be exhaustive or to limit the present invention tothe forms disclosed. Accordingly, many modifications and variations willbe apparent to practitioners skilled in the art. Additionally, the abovedisclosure is not intended to limit the present invention. The scope ofthe present invention is defined by the appended claims.

1. A computer-implemented method for determining a reconstructed maskpattern, comprising: receiving a target mask pattern; receiving an imageof at least a portion of a photo-mask corresponding to a mask pattern,wherein the image is determined using inspection optics having anoptical path; and reconstructing, using a computer, the mask patternfrom the image based on a characteristic of the optical path and thetarget mask pattern, wherein the mask pattern is reconstructed using aconstrained inverse optical calculation in which there are a finitenumber of discrete feature widths allowed in the reconstructed maskpattern, and wherein a given feature has a constant feature width. 2.The method of claim 1, wherein the reconstructed mask pattern ischaracterized by additional spatial frequencies than the image.
 3. Themethod of claim 1, wherein, in the constrained inverse opticalcalculation, the image is at an image plane of a model of the opticalpath and the mask pattern is at an object plane of the model of theoptical path.
 4. The method of claim 1, wherein the characteristicincludes a range of wavelengths transmitted via the optical path.
 5. Themethod of claim 1, wherein the features in the reconstructed maskpattern each have the constant feature width.
 6. The method of claim 5,wherein the constant feature width is an average critical dimension ofthe reconstructed mask pattern.
 7. The method of claim 6, wherein themethod further comprises comparing the average critical dimension of thereconstructed mask pattern to an average critical dimension of thetarget mask pattern to determine a critical-dimension bias of thephoto-mask.
 8. The method of claim 7, wherein the method furthercomprises determining an acceptance condition of the photo-mask based onthe critical-dimension bias.
 9. The method of claim 6, wherein theaverage critical dimension is associated with a minimum of a costfunction in the inverse optical calculation; and wherein the costfunction corresponds to a difference between the reconstructed maskpattern and the target mask pattern.
 10. The method of claim 1, wherein,by constraining the inverse optical calculation a convergence time ofthe inverse optical calculation is faster than that of an unconstrainedinverse optical calculation.
 11. The method of claim 1, wherein primaryfeatures in the reconstructed mask pattern each have the constantfeature width; and wherein sub-resolution assist features in thereconstructed mask pattern each have another constant feature width,which is different than the constant feature width.
 12. The method ofclaim 1, wherein the spatial frequencies in the image are greater thanor equal to a spatial frequency corresponding to an average criticaldimension of the target mask pattern.
 13. A non-transitorycomputer-program product for use in conjunction with a computer system,the computer-program product comprising a computer-readable storagemedium and a computer-program mechanism embedded therein to determine areconstructed mask pattern, the computer-program mechanism including:instructions for receiving a target mask pattern; instructions forreceiving an image of at least a portion of a photo-mask correspondingto a mask pattern, wherein the image is determined using inspectionoptics having an optical path; and instructions for reconstructing themask pattern from the image based on a characteristic of the opticalpath and the target mask pattern, wherein the mask pattern isreconstructed using a constrained inverse optical calculation in whichthere are a finite number of discrete feature widths allowed in thereconstructed mask pattern, and wherein a given feature has a constantfeature width.
 14. The computer-program product of claim 13, wherein thereconstructed mask pattern is characterized by additional spatialfrequencies than the image.
 15. The computer-program product of claim13, wherein, in the constrained inverse optical calculation, the imageis at an image plane of a model of the optical path and the mask patternis at an object plane of the model of the optical path.
 16. Thecomputer-program product of claim 13, wherein the characteristicincludes a range of wavelengths transmitted via the optical path. 17.The computer-program product of claim 13, wherein the features in thereconstructed mask pattern each have the constant feature width.
 18. Thecomputer-program product of claim 17, wherein the constant feature widthis an average critical dimension of the reconstructed mask pattern. 19.The computer-program product of claim 18, wherein the computer-programproduct further includes comparing the average critical dimension of thereconstructed mask pattern to an average critical dimension of thetarget mask pattern to determine a critical-dimension bias of thephoto-mask.
 20. The computer-program product of claim 19, wherein thecomputer-program product further includes determining an acceptancecondition of the photo-mask based on the critical-dimension bias. 21.The computer-program product of claim 18, wherein the average criticaldimension is associated with a minimum of a cost function in the inverseoptical calculation; and wherein the cost function corresponds to adifference between the reconstructed mask pattern and the target maskpattern.
 22. The computer-program product of claim 18, wherein, byconstraining the inverse optical calculation a convergence time of theinverse optical calculation is faster than that of an unconstrainedinverse optical calculation.
 23. The computer-program product of claim18, wherein primary features in the reconstructed mask pattern each havethe constant feature width; and wherein sub-resolution assist featuresin the reconstructed mask pattern each have another constant featurewidth, which is different than the constant feature width.
 24. Thecomputer-program product of claim 18, wherein the spatial frequencies inthe image are greater than or equal to a spatial frequency correspondingto an average critical dimension of the target mask pattern.
 25. Acomputer system, comprising: at least one processor; at least onememory; and at least one program module, the program module stored inthe memory and configured to be executed by the processor to determine areconstructed mask pattern, the program module including: instructionsfor receiving a target mask pattern; instructions for receiving an imageof at least a portion of a photo-mask corresponding to a mask pattern,wherein the image is determined using inspection optics having anoptical path; and instructions for reconstructing the mask pattern fromthe image based on a characteristic of the optical path and the targetmask pattern, wherein the mask pattern is reconstructed using aconstrained inverse optical calculation in which there are a finitenumber of discrete feature widths allowed in the reconstructed maskpattern, and wherein a given feature has a constant feature width.